Nonequilibrium Fluctuation-dissipation Relation Research Articles

Thermalization with a multibath: an investigation in simple models

We study analytically and numerically a couple of paradigmatic spin models, each described in terms of two sets of variables attached to two different thermal baths with characteristic timescales T and τ and inverse temperatures B and β. In the limit in which one bath becomes extremely slow (), such models amount to a paramagnet and to a one-dimensional ferromagnet in contact with a single bath. Our study is also motivated by analogies with disordered systems where widely separated timescales associated with different effective temperatures emerge. We show that these systems reach a stationary state in a finite time for any choice of B and β. We determine the non-equilibrium fluctuation-dissipation relation between the autocorrelation and the response function in such a state and, from that, we discuss if and how thermalization with the two baths occurs and the emergence of a non-trivial fluctuation-dissipation ratio.

Predicting brain evoked response to external stimuli from temporal correlations of spontaneous activity

The relation between spontaneous and stimulated global brain activity is a fundamental problem in the understanding of brain functions. This question is investigated both theoretically and experimentally within the context of nonequilibrium fluctuation-dissipation relations. We consider the stochastic coarse-grained Wilson-Cowan model in the linear noise approximation and compare analytical results to experimental data from magnetoencephalography (MEG) of human brain. The short time behavior of the autocorrelation function for spontaneous activity is characterized by a double-exponential decay, with two characteristic times, differing by two orders of magnitude. Conversely, the response function exhibits a single exponential decay in agreement with experimental data for evoked activity under visual stimulation. Results suggest that the brain response to weak external stimuli can be predicted from the observation of spontaneous activity and pave the way to controlled experiments on the brain response under different external perturbations.

Fluctuation-dissipation relation and stationary distribution of an exactly solvable many-particle model for active biomatter far from equilibrium.

An exactly solvable, Hamiltonian-based model of many massive particles that are coupled by harmonic potentials and driven by stochastic non-equilibrium forces is introduced. The stationary distribution and the fluctuation-dissipation relation are derived in closed form for the general non-equilibrium case. Deviations from equilibrium are on one hand characterized by the difference of the obtained stationary distribution from the Boltzmann distribution; this is possible because the model derives from a particle Hamiltonian. On the other hand, the difference between the obtained non-equilibrium fluctuation-dissipation relation and the standard equilibrium fluctuation-dissipation theorem allows us to quantify non-equilibrium in an alternative fashion. Both indicators of non-equilibrium behavior, i.e., deviations from the Boltzmann distribution and deviations from the equilibrium fluctuation-dissipation theorem, can be expressed in terms of a single non-equilibrium parameter α that involves the ratio of friction coefficients and random force strengths. The concept of a non-equilibrium effective temperature, which can be defined by the relation between fluctuations and the dissipation, is by comparison with the exactly derived stationary distribution shown not to hold, even if the effective temperature is made frequency dependent. The analysis is not confined to close-to-equilibrium situations but rather is exact and thus holds for arbitrarily large deviations from equilibrium. Also, the suggested harmonic model can be obtained from non-linear mechanical network systems by an expansion in terms of suitably chosen deviatory coordinates; the obtained results should thus be quite general. This is demonstrated by comparison of the derived non-equilibrium fluctuation dissipation relation with experimental data on actin networks that are driven out of equilibrium by energy-consuming protein motors. The comparison is excellent and allows us to extract the non-equilibrium parameter α from experimental spectral response and fluctuation data.

Spatial dispersion in atom-surface quantum friction

We investigate the influence of spatial dispersion on atom-surface quantum friction. We show that for atom-surface separations shorter than the carrier's mean free path within the material, the frictional force can be several orders of magnitude larger than that predicted by local optics. In addition, when taking into account spatial dispersion effects, we show that the commonly used local thermal equilibrium approximation underestimates by approximately 95% the drag force, obtained by employing the recently reported nonequilibrium fluctuation-dissipation relation for quantum friction. Unlike the treatment based on local optics, spatial dispersion in conjunction with corrections to local thermal equilibrium not only change the magnitude but also the distance scaling of quantum friction.

Turbulence as a Problem in Non-equilibrium Statistical Mechanics

The transitional and well-developed regimes of turbulent shear flows exhibit a variety of remarkable scaling laws that are only now beginning to be systematically studied and understood. In the first part of this article, we summarize recent progress in understanding the friction factor of turbulent flows in rough pipes and quasi-two-dimensional soap films, showing how the data obey a two-parameter scaling law known as roughness-induced criticality, and exhibit power-law scaling of friction factor with Reynolds number that depends on the precise form of the nature of the turbulent cascade. These results hint at a non-equilibrium fluctuation-dissipation relation that applies to turbulent flows. The second part of this article concerns the lifetime statistics in smooth pipes around the transition, showing how the remarkable super-exponential scaling with Reynolds number reflects deep connections between large deviation theory, extreme value statistics, directed percolation and the onset of coexistence in predator-prey ecosystems. Both these phenomena reflect the way in which turbulence can be fruitfully approached as a problem in non-equilibrium statistical mechanics.

Perturbative fluctuation dissipation relation for nonequilibrium finite-frequency noise in quantum circuits

We develop a general perturbative computation of finite-frequency quantum noise which applies, in particular, to both good or weakly transmitting strongly correlated conductors coupled to a generic environment. Under a minimal set of hypotheses, we show that the noise can be expressed through the non-equilibrium DC current only, generalizing a non-equilibrium fluctuation dissipation relation. We use this relation to derive explicit predictions for the non equilibrium finite frequency noise for a single channel conductor connected to an arbitrary Ohmic environment.

Quantum friction and fluctuation theorems

We use general concepts of statistical mechanics to compute the quantum frictional force on an atom moving at constant velocity above a planar surface. We derive the zero-temperature frictional force using a non-equilibrium fluctuation-dissipation relation, and show that in the large-time, steady-state regime quantum friction scales as the cubic power of the atom's velocity. We also discuss how approaches based on Wigner-Weisskopf and quantum regression approximations fail to predict the correct steady-state zero temperature frictional force, mainly due to the low frequency nature of quantum friction.

Nonequilibrium fluctuation-dissipation relations for one- and two-particle correlation functions in steady-state quantum transport.

We study the non-equilibrium (NE) fluctuation-dissipation (FD) relations in the context of quantum thermoelectric transport through a two-terminal nanodevice in the steady-state. The FD relations for the one- and two-particle correlation functions are derived for a model of the central region consisting of a single electron level. Explicit expressions for the FD relations of the Green's functions (one-particle correlations) are provided. The FD relations for the current-current and charge-charge (two-particle) correlations are calculated numerically. We use self-consistent NE Green's functions calculations to treat the system in the absence and in the presence of interaction (electron-phonon) in the central region. We show that, for this model, there is no single universal FD theorem for the NE steady state. There are different FD relations for each different class of problems. We find that the FD relations for the one-particle correlation function are strongly dependent on both the NE conditions and the interactions, while the FD relations of the current-current correlation function are much less dependent on the interaction. The latter property suggests interesting applications for single-molecule and other nanoscale transport experiments.

Nonequilibrium fluctuation-dissipation inequality and nonequilibrium uncertainty principle

The fluctuation-dissipation relation is usually formulated for a system interacting with a heat bath at finite temperature, and often in the context of linear response theory, where only small deviations from the mean are considered. We show that for an open quantum system interacting with a nonequilibrium environment, where temperature is no longer a valid notion, a fluctuation-dissipation inequality exists. Instead of being proportional, quantum fluctuations are bounded below by quantum dissipation, whereas classically the fluctuations vanish at zero temperature. The lower bound of this inequality is exactly satisfied by (zero-temperature) quantum noise and is in accord with the Heisenberg uncertainty principle, in both its microscopic origins and its influence upon systems. Moreover, it is shown that there is a coupling-dependent nonequilibrium fluctuation-dissipation relation that determines the nonequilibrium uncertainty relation of linear systems in the weak-damping limit.

Nonequilibrium fluctuation-dissipation relation from holography

We derive a non-equilibrium fluctuation-dissipation relation for bosonic correlation functions from holography in the classical gravity approximation at strong coupling. This generalizes the familiar thermal fluctuation-dissipation relation in absence of external sources. This also holds universally for any non-equilibrium state which can be obtained from a stable thermal equilibrium state in perturbative derivative (hydrodynamic) and amplitude (non-hydrodynamic) expansions. Therefore, this can provide a strong experimental test for the applicability of the holographic framework. We discuss how it can be tested in heavy ion collisions. We also make a conjecture regarding multi-point holographic non-equilibrium Green's functions.

Fluctuation-Dissipation Relations in Minimal Models for Active Driving

Biological systems exhibit fluctuations driven by active (energy consuming) processes, such as molecular motors in the cytoskeleton and flagella propelling bacteria. It is tempting to treat these fluctuations in the same manner that thermal fluctuations are treated in equilibrium systems. Recent experiments have compared these inherent fluctuations with the response to small external perturbations. From these measurements, an effective temperature may be defined using the fluctuation-dissipation formalism of equilibrium statistical mechanics. In active systems, this effective temperature is frequency dependent (unlike the thermal case), and is higher than the ambient temperature. Another parameter that is useful for characterizing deviations from equilibrium of such fluctuations is the kurtosis of their distribution function; deviations of the kurtosis from its value for a Gaussian distribution indicate that the system may not be mapped onto an equilibrium ensemble.We address these issues theoretically by examining simple model systems of randomly kicked “particles”, with a variable number of kicking “motors”. Our generalized particles and motors represent different objects in different real systems, and we make specific choices when comparing to experimental observations. We exactly calculate the non-equilibrium fluctuation-dissipation relations and resulting effective temperatures. We show that their frequency dependence is model dependent and generally non-monotonous. We investigate the non-Gaussianity of the velocity distribution functions by a combination of numerical simulations and theoretical estimates. We show that the kurtosis has a non-monotonous dependence on the motor activity. We identify situations in which these two indicators of non-equilibrium behavior contradict. In particular, in the presence of multiple motors, the kurtosis of the distribution recovers its Gaussian value, while the effective temperature may still exhibit strong frequency dependence. We compare our results with two recent measurements of oscillations of the red-blood cell membrane, where these two indicators where independently measured.

Nonequilibrium fluctuation-dissipation relations of interacting Brownian particles driven by shear

We present a detailed analysis of the fluctuation-dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At short times, the equilibrium FDT holds. We follow two independent approaches whose results are in qualitative agreement. To discuss the derived fluctuation-dissipation ratios, we show an exact reformulation of the susceptibility which contains not the full Smoluchowski operator as in equilibrium, but only its well defined Hermitian part. This Hermitian part can be interpreted as governing the dynamics in the frame comoving with the probability current. We present a simple toy model which illustrates the FDT violation in the sheared colloidal system.

Admittance and Noise in an Electrically Driven Nanostructure: Interplay between Quantum Coherence and Statistics

We investigate the interplay between the quantum coherence and statistics in electrically driven nanostructures. We obtain an expression for the admittance and the current noise for a driven nanocapacitor in terms of the Floquet scattering matrix and derive a nonequilibrium fluctuation-dissipation relation. As an interplay between the quantum phase coherence and the many-body correlation, the admittance has peak values whenever the noise power shows a step as a function of the nearby gate voltage. Our theory is demonstrated by calculating the admittance and noise of driven double-quantum dots.

Probing a Nonequilibrium Einstein Relation in an Aging Colloidal Glass

We present a direct experimental measurement of an effective temperature in a colloidal glass of laponite, using a micrometric bead as a thermometer. The nonequilibrium fluctuation-dissipation relation, in the particular form of a modified Einstein relation, is investigated with diffusion and mobility measurements of the bead embedded in the glass. We observe an unusual nonmonotonic behavior of the effective temperature: starting from the bath temperature, it is found to increase up to a maximum value, and then decrease back, as the system ages. We show that the observed deviation from the Einstein relation is related to the relaxation times previously measured in dynamic light scattering experiments.

Non-equilibrium fluctuation—dissipation relations in binary glasses

We show that at short time aging is realized in a simple model of binary glasses. We find that below the transition point the off-equilibrium correlation functions and response functions are compatible with the relations that were originally derived Cugliandolo Kurchan for generalized spin glasses.

Non-equilibrium fluctuation-dissipation relations in binary glasses

Non-equilibrium fluctuation-dissipation relations in binary glasses

Nonequilibrium fluctuation–dissipation relations for independent random rate processes with dynamical disorder

A class of rate processes with dynamical disorder is investigated based on the two following assumptions: (a) the system is composed of a random number of particles (or quasiparticles) which decay according to a first-order kinetic law; (b) the rate coefficient of the process is a random function of time with known stochastic properties. The formalism of characteristic functionals is used for the direct computation of the dynamical averages. The suggested approach is more general than the other approaches used in the literature: it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient. We derive an infinity of exact fluctuation–dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles. The analysis of these fluctuation–dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of particles have an intermittent behavior. The moments are explicitly evaluated in two particular cases: (a) the random behavior of the rate coefficient is given by a non-Markovian process which can be embedded in a Markovian process by increasing the number of state variables and (b) the stochastic behavior of the rate coefficient is described by a stationary Gaussian random process which is generally non-Markovian. The method of curtailed characteristic functionals is used to recover the conventional description of dynamical disorder in terms of the Kubo–Zwanzig stochastic Liouville equations as a particular case of our general approach. The fluctuation–dissipation relations can be used for the study of fluctuations without making use of the whole mathematical formalism. To illustrate the efficiency of our method for the analysis of fluctuations we discuss three different physicochemical and biochemical problems. A first application is the kinetic study of the decay of positrons or positronium atoms thermalized in dense fluids: in this case the time dependence of the rate coefficient is described by a stationary Gaussian random function with an exponentially decaying correlation coefficient. A second application is an extension of Zwanzig’s model of ligand–protein interactions described in terms of the passage through a fluctuating bottle neck; we complete the Zwanzig’s analysis by studying the concentration fluctuations. The last example deals with jump rate processes described in terms of two independent random frequencies; this model is of interest in the study of dielectric or conformational relaxation in condensed matter and on the other hand gives an alternative approach to the problem of protein–ligand interactions. We evaluate the average survival function in several particular cases for which the jump dynamics is described by two activated processes with random energy barriers. Depending on the distributions of the energy barriers the average survival function is a simple exponential, a stretched exponential, or a statistical fractal of the inverse power law type. The possible applications of the method in the field of biological population dynamics are also investigated.

A variation principle in the quantum theory of irreversible processes formulated in a superspace

A method is proposed for investigating the nonequilibrium fluctuations in an arbitrary system subjected to the influence of an external perturbation. We reformulate Nakano's variation principle in a Liouville (super-) space. An exact formula is obtained for the entropy production of a scattering problem. The formula is a matrix element of the scattering super-operator. An exact formula is also given for a nonlinear nonequilibrium fluctuation-dissipation relation.